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Revision Notes on Unit and Dimensions

In order to make the measurement of a physical quantity we have, first of all, to evolve a standard for that measurement so that different measurements of same physical quantity can be expressed relative to each other. That standard is called a unit of that physical quantity.

  • System of Units:-

(a) C.G.S (Centimeter-Grand-Second) system.

(b) F.P.S. (Foot-Pound-Second) system.

(c) M.K.S. (Meter-Kilogram--Second) system.

(d) M.K.S.A. (Meter-Kilogram-Second-Ampere) unit.

  • Dimensional Formula:-

Dimensional formula of a physical quantity is the formula which tells us how and which of the fundamental units have been used for the measurement of that quantity.

  • How to write dimensions of physical quantities:-

(a) Write the formula for that quantity, with the quantity on L.H.S. of the equation.

(b) Convert all the quantities on R.H.S. into the fundamental quantities mass, length and time.

(c) Substitute M, L and T for mass, length and time respectively.

(d) Collect terms of M,L and T and find their resultant powers (a,b,c) which give the dimensions of the quantity in mass, length and time respectively.

  • Characteristics of Dimensions:-

(a) Dimensions of a physical quantity are independent of the system of units.

(b) Quantities having similar dimensions can be added to or subtracted from each other.

(c) Dimensions of a physical quantity can be obtained from its units and vice-versa.

(d) Two different physical quantities may have same dimensions.

(e) Multiplication/division of dimensions of two physical quantities (may be same or different) results in production of dimensions of a third quantity.

PHYSICAL QUANTITY

SYMBOL

DIMENSION

MEASUREMENT UNIT

UNIT

Length

s

L

Meter

m

Mass

M

Kilogram

Kg

Time

t

T

Second

Sec

Electric charge

q

Q

Coulomb

C

luminous intensity

I

C

Candela

Cd

Temperature

T

K

Kelvin

oK

Angle

 

q

none

Radian

None

Mechanical Physical Quantities (derived)

 

PHYSICAL QUANTITY

SYMBOL

DIMENSION

MEASUREMENT  UNIT

UNIT

 

 

 

 

 

Area

A

L2

square meter

m2

Volume

 

V

L3

cubic meter 

m3

velocity

 

v

L/T

meter per second

m/sec

angular velocity

 

w

T-1

radians per second

1/sec

acceleration

 

a

LT-2

meter per square second

m/sec2

angular acceleration

a

T-2

radians per square

second 

1/sec2

Force

 

F

MLT-2

 Newton

Kg m/sec2

Energy

 

E

ML2T-2

 Joule

Kg m2/sec2

Work

 

W

       

ML2T-2

Joule

Kg m2/sec2

Heat

 

Q

     

ML2T-2

Joule

Kg m2/sec2

Torque

 

t

ML2T-2

Newton meter

Kg m2/sec2

Power

 

P

ML2T-3

watt  or  joule/sec

Kg m2/sec3

Density

 

D or ρ

ML-3

kilogram per

cubic meter

Kg/m3

pressure

 

P

    ML-1T-2

Newton per square meter

Kg m-1/sec2

impulse

 

J

MLT-1

Newton second

Kg m/sec

Inertia

 

I

ML2

Kilogram square meter

 

Kg m2 

 

luminous 

flux

f

 

C

lumen (4Pi candle for point source)

cd sr 

illumination

 

E

CL-2

lumen per

square meter

cd sr/m2

entropy

 

S

       ML2T-2K-1

joule per degree

Kg m2/sec2K

Volume

rate of flow

Q

L3T-1

cubic meter

per second

m3/sec

kinematic

viscosity

n

L2T-1

square meter

m2/sec

per second

 

dynamic

viscosity

m

 

      ML-1T-1

 

Newton second

per square meter

Kg/m sec        

 

specific

weight

g

  ML-2T-2

Newton

per cubic meter

Kg m-2/sec2

Electrical Physical Quantities (derived)

Electric

current

I

QT-1

Ampere

C/sec

emf, voltage,

potential

E

ML2T-2Q-1

Volt

Kg m2/sec2C

resistance or 

impedance 

R

 

ML2T-1Q-2

 

ohm

 

Kgm2 /secC2

 

Electric 

conductivity

s

M-2L-2TQ2

mho 

 

secC2/Kg m3 

 

capacitance

 

C

M-1L-2T2 Q2

Farad

sec2C2/Kgm2

inductance

 

L

ML2Q-2

Henry

Kg m2 /C2

Current density

 

J

 

QT-1L-2

 

ampere per

square meter

C/sec m2

 

Charge density

r

QL-3

coulomb per cubic meter

C/m3

magnetic flux,

Magnetic induction

B

MT-1Q-1

weber per

square meter

Kg/sec C

magnetic

intensity

H

QL-1T-1

ampere per meter

C/m sec

magnetic vector

potential

A

MLT-1Q-1

weber/meter

Kg m/sec C

Electric

field intensity

E

MLT-2Q-1

volt/meter or

newton/coulomb

Kg m/sec2 C

Electric displacement

D

QL-2

coulomb per square meter

C/m2

permeability

 

m

MLQ-2

henry per meter

Kg m/C2

permittivity,

e

T2Q2M-1L-3

farad per meter

sec2C2/Kgm3

 

dielectric constant

                     

K

    M0L0T0

 None

 None

frequency

 

f or n

T-1

Hertz

sec-1

angular frequency

W

T-1

radians per second         

sec-1

 

Wave length

 

l

L

Meters

M

 

  • Principle of homogeneity:-

It states that “ the dimensional formulae of every term on the two sides of a correct relation must be same.”

  • Types of error:-

(a) Constant errors:- An error is said to be constant error if it affects, every time, a measurement in a similar manner.

(b) Systematic errors:- Errors which come into existence by virtue of a definite rule, are called systematic errors.

(c) Random error or accidental error:- Error which takes place in a random manner and cannot be associated with a systematic cause are called random or accidental errors.

(d) Absolute error:-  \Delta x_{i} = x_{i} - \bar{x}  

  • Relative Error:-

 \delta x_{r} = \frac{\bar{\Delta x}}{\bar{x}}

  • Percentage Error:-

\delta x_{p} = \frac{\bar{\Delta x}}{\bar{x}}\times 100

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